Lattices can either be primitive (only one lattice point per unit cell) or
non-primitive (more than one lattice point per unit cell).

If you combine the 7 crystal systems with the 2 different types of lattices, you
end up with 14 Bravais Lattices (named after Auguste Bravais who figured all this out in
1850).

Another way to answer your question is to categorize crystals by their
physical/chemical properties. In this classification you have four types of crystals:

Covalent Crystals:

This is a crystal which has real chemical covalent between all of the atoms in
the crystal. So really a single crystal of a covalent crystals is really just one big
molecule. An example of this is a crystal like diamond or zinc sulfide. Covalent crystals
can have extremely high melting points.

Metallic Crystals:

Individual metal atoms sit on lattice sites while the outer electrons from these
atoms are able to flow freely around the lattice. Metallic crystals normally have high
melting points and densities.

Ionic Crystals:

This is a crystal where the individual atoms don't have covalent bonds between
them, but are held together by electrostatic forces. An example of this type of crystal is
sodium chloride (NaCl). Ionic crystals are hard and have relatively high melting points.

Molecular Crystals:

This is a crystal where there are recognizable molecules in the structure and the
crystal is held together by non-covalent interactions like van der Waals forces or
hydrogen bonding. An example of this type of crystal would be sugar. Molecular crystals
tend to be soft and have lower melting points.